According to Maxwell equations, steady currents and steady charge density won't produce EM waves. So if you have a steady current loop, you can calculate its magnetic field just by the Biot-Savart law, which gives an steady magnetic field in space. If a current loop would radiate energy, then it would be impossible to produce persistent currents in superconductor loops which are actually experimentally observed.(Actually there is also the quantum mechanics in play, and using "only" the classical Maxwell theory isn't completely correct, but it gives you the idea)
But to explain your point about a loop current being charges moving in circle, you are partially correct. If you break the current into tiny elements and calculate the electromagnetic field of each individual element, surely you see that the EM field of each individual element is a propagating magnetic field, but when you sum the EM fields of various elements together using superposition principle, the resulting "net electro magnetic field" would be the same as the one that is given by the Biot-Savart law. (Somehow the same that happen in interference experiments.)
But there is two points to make; Although the steady current and zero charge density is theoretically presumable, this could not be the case in the real world. Since the currents are moving electrons you expect that if you zoom in enough, you see single electrons with some gaps between them, So it seems that in that scale the charge density isn't steady in time and the continuous approximation of charge density breaks in that scale.(Although the currents are produced by electrons in the Bloch states which are not localized in space, So the moving single electrons picture isn't entirely correct either). And if you have a time-varying charge density, you would get EM radiation. Actually this is the situation in the synchrotrons. There you don't have an steady uniform current around the loop, but a bunch of individual electrons that are circulating the loop and you can see the time varying charge density easily. You just can't approximate the electrons running through the loop by a steady current.